3. MULTI-QUADRATIC DYNAMIC PROGRAMMING PROCEDURE FOR IMAGE PROCESSING
In this paper, we propose a new non-convex type of pair-wise potential functions, allows more flexibility to set a priori
preferences, using different coefficients of penalty for various ranges of differences between the values of adjacent image
elements:
1 1
1 1
, min
, , , ,
L
x x x x
x x
t t t t t t
t t t
6
The developed image analysis procedure can significantly extend the class to solve applied problems, to take into account
the presence of heterogeneities and discontinuities in the original data, while retaining the high computational efficiency
of procedures of dynamic programming and Kalman filter- interpolator Kalman and Bucy, 1961.
It can be proven, that if the pair-wise Gibbs potentials are selected as a minimum of a finite set of quadratic functions, and
node functions are in quadratic form, the procedure breaks down at each step into several parallel procedures, according to
the number of quadratic functions forming the intermediate optimization problems of one variable. The Bellman function at
each step of the dynamic programming will has a minimum of a finite set of quadratic functions:
1 2
min , ,...,
L
J x J x
J x
J x
t t t t
t t
t t
7 where
2
, 0, 1,..,
i i
i i
i
J x
q x x
d q i
L
t t
t t t
t t
.
8 Let
1
1 1
1
min ,
x
F x x
x J
x
t
t t t
t t
t t
then
1 1
1 1
1 1
1 1
1 1
1 1
1
min ,
, min
min ,
t x
L x
x x
x F
x F x
x x
x F
x
t t
t t t
t t
t t t
t t
t t
t t
t
9
1 2
min ,
, ,
K
F x F x
F x
F x
t t t
t t
t t
t
10
2
, 0,
1,..,
j j
j j
j
F x
q x
x d
q j
K
t t
t t
t t
t
11
The backward recurrent relation
1
ˆ x
x
t
t
has the following form:
1 1
1
ˆ argmin
t t
t t
x x
F x x
12
1 1
1 1
1 1
min ,
t t t
t t
t t
t
j i
F x x
x x
F x
13 It is easy to see that if the values , ,..,
x x x
1 2
N
are the minimum points of criteria 1, then
, x
a b
t
for every
1,..,
t N , where
min a
Y
and min
a Y
. It can be noted that not all of the
quadratic function will participate in forming of the final function, because their values are not minimum for any point.
Such functions can be dropped using enough simple procedure that takes into account the position of the minimum point and
the points of intersection of quadratic functions with each other. At the beginning, sort by ascending values
j
d
t
of array
j
F
t
. At each step, looking for the minimum constant and discard all
others constant. Discard all functions than have minimum greater than or equal to this constant.
After that, we find necessary and sufficient condition of intersection of quadratic function by following equations:
2 2
i i
i j
j j
q x
x d
q x
x d
t t
t t
t t
t t
14 where ,
1.. ; i j
K i j
The coordinates of the intersection points on the real axis defined by the following relations:
1
_ 1
i j
ij i
i j
j q
q
x c q
x q
x
t t
t t
t t
t
15
1
_ 2
i j
ij i
i j
j q
q
x c q
x q
x
t t
t t
t t
t
16 The points of intersection
_ 1
ij
x c
t
and _ 1
ij
x c
t
have real coordinates, if expression
under the square root is greater than or equal to zero:
2 i
j j
i j
i i
j
q q
x x
d d
q q
t t
t t
t t
t t
17 After that, select functions that have points of intersection with
satisfying _ 1 , _ 2
ij ij
a x c
x c b
t t
. Discard all the functions for which there is no intersection. Among the tested functions,
select the function with the smallest minimum and leave it. Check its intersection with other functions. Discard all
functions for which there is no intersection. Repeat as long as there will be functions for which have no decision on
acceptance or discarding. Then the reduction of the amount Bellman functions at each step is determined by the formula:
1 2
min ,
, , ,
H
F x F x
F x
F x
H K
t t t
t t
t t
t
18
The number of Bellman functions can be reduced according to the expression 17. It was proven, that in the case of signal
processing the number of quadratic functions that are required for representation of a Bellman function, generally, does not
increase by more than one at each step. Nevertheless, using this approach for processing data based on the two-dimensional the
tree approximation of lattice graph neighborhood Mottl et.al, 1998, the number of quadratic functions on the Bellman
functions may be too large and leads to a lack of effective implementation of the procedure.
The basic idea of the proposed procedure is to find the groups of closest, in the appropriate sense, quadratic functions using k-
means clustering algorithm, and to replace each of these groups by one quadratic function having the smallest minimun value.
We consider quadratic functions in 17 as a points in three- dimensional space with coordinates
, , , 1,...,
i i
i
q x d i
H
t t t
. Using k-means clustering algorithm, the distance between
quadratic functions can be calculated as the sum of the squares of the difference on the definitive range [a, b]:
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XL-5-W6-101-2015
103
2 2
t t
t t
t t
t t
t
b i
i i
j j
j ij
a
D q
x x
d q
x x
d dx 19
2 2
2 1 2
2 3 3
1 2 3
4 1 3 5 1 2
2
ij
D c s
c s c
s s s
c s s c s s
20 where
5 5
1
5 b
a c
;
2
c b
a ;
3 3
3
2 3
b a
c
;
2 2
4
2 c
a b
;
4 4
5
c a
b
;
1 i i
j j
s q x
q x
t t t t
;
2 i
j
s q
q
t t
;
2 2
3 i
i j
j i
j
s q x
q x d
d
t t t t
t t
; ; ,
1,..., i
j i j H
Assume that the number k is a predetermined number of groups for k-means clustering algorithm. To preserve the
quality of image processing, we do not use directly the final cluster centers which derived by the k-means algorithm. For
each of the derived final groups we choose a point that have the lowest third coordinate.
Using the algorithm for reduction in the number of quadratic functions allows you to get the effective implementation of
image processing procedure on the basis multi quadratic dynamic programming procedure. Experimental studies show
that the vast majority of the original data sets of two or three square functions that are quite fully reflect each function in the
Bellman criteria 7.
4. EXPERIMENTAL RESULTS
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